CalculationofInverseLaplaceTransformProcedureoutlinedforrational functionsofs.G(s) -b*+b-bs+h _ ()s"+a-s"--+.as+aoP(s)P0=iG-A)UNon-repeated linear factors:之,k,=m[(s-~),G()] =...nG(s)=)Bs-AIn effect:=Lfe'),e C, 1≥0= g(0)=2k-exp(an),12011Calculationof InverseLaplaceTransformProcedureoutlinedforrational functionsof s.G(s)=b+b+bs+h_2()s"+a.-s"+.as+aoP(s)In the case that the characteristic polynomial obtains a pair of complex conjugateroots, Ae,, the factorized form includes the quadratic factor(s-)-(s-)=2 -2 Re()-$+)Ineffect:kc+ko→ke.=(ke)-2.Re()s+--126
6 11 Calculation of Inverse Laplace Transform ( ) 1 1 1 0 1 1 1 0 ( ) ( ) m m m m n n n b s b s b s b Q s G s s a s a s a P s − − − − + + + = = + + + Non-repeated linear factors: [ ] 1 1 ( ) ( ) ( ) , lim ( ) ( ) , 1,., v n v v n v v v s v v P s s k G s k s G s v n s λ Π λ λ λ = → = = − ⇓ = = − ⋅ = − ∑ In effect: { } 1 1 , , 0 ( ) exp( ), 0 n t v v v e t g t k t t s λ λ λ λ = = ∈ ≥ ⇒ = ⋅ ≥ − L ∑ Procedure outlined for rational functions of s. 12 Calculation of Inverse Laplace Transform ( ) 1 1 1 0 1 1 1 0 ( ) ( ) m m m m n n n b s b s b s b Q s G s s a s a s a P s − − − − + + + = = + + + In the case that the characteristic polynomial obtains a pair of complex conjugate roots, , λ λ C C ∗ , the factorized form includes the quadratic factor 2 2 ( ) ( ) 2 Re( ) C C C C s s s s λ λ λ λ ∗ − ⋅ − = − ⋅ ⋅ + In effect: ( ) 2 2 1 2 Re( ) C C C C C C C C k k k k s s s s λ λ λ λ ∗ ∗ = + ∗ ⇒ ∗ = − ⋅ ⋅ + − − Procedure outlined for rational functions of s.
Calculation of Inverse Laplace TransformProcedure outlined for rational functions of s.G(s)=-b+b-bs+h_2()s"+a.-s"-l +..a,s+aoP(s)Repeated linear factors:P(0)=(s-2) (s-2)= G(s)=2+2_k0台s-台(s-）di-μ(-)[s-) G] = koμ=lim-1).4In effect:(u-1)13Solutionof LinearODEswithconstantcoefficientsNon-homogeneous equation with constant coefficientsL[y](t)= f(t)↑a,y("(t)+an--y(n-'(t)+...+ay(t)+aoy(t)=f(t),a,+0Initial conditionsy(to) = %, y (to) = Yi..,y(-l(t) = Y-!The Laplace Transform can be used for the both the general and theparticular solution.147
7 13 Calculation of Inverse Laplace Transform ( ) 1 1 1 0 1 1 1 0 ( ) ( ) m m m m n n n b s b s b s b Q s G s s a s a s a P s − − − − + + + = = + + + Repeated linear factors: ( ) ( ) ( ) ( ) ( ) 0 0 0 1 1 1 0 0 0 ( ) ( ) ( ) , 1 lim ( ) ( ) , 1,., ! n n l l v v v v v l l l s k k P s s s G s s s d k s G s l l ds µ µ µ µ µ µ λ λ Π λ λ λ λ µ µ = = = − → − = − − ⇒ = + − − = ⋅ − ⋅ = − ∑ ∑ In effect: ( ) ( ) ( ) 0 1 1 0 1 1 1 ! t t e s µ λ µ λ µ − − = − − L Procedure outlined for rational functions of s. 14 Solution of Linear ODEs with constant coefficients [ ] ( ) ( 1) (1) 1 1 0 ( ) ( ) ( ) ( ) ( ) ( ) ( ), 0 n n n n n L y t f t a y t a y t a y t a y t f t a − − = + + + + = ≠ Non-homogeneous equation with constant coefficients The Laplace Transform can be used for the both the general and the particular solution. ( 1) 0 0 0 1 0 1 ( ) , '( ) , , ( ) n n y t y t y t γ γ γ − = = = . − Initial conditions